Question

The wavelength of light observed on earth from a moving star is found to be decreased by $0.5\%$. Relative to earth star is:
A. Moving away with velocity $1.5 \times {10^5}\dfrac{m}{s}$
B. Moving away with velocity $1.5 \times {10^4}\dfrac{m}{s}$
C. Moving closer with velocity $1.5 \times {10^5}\dfrac{m}{s}$
D. Moving closer with velocity $1.5 \times {10^4}\dfrac{m}{s}$

Answer 1

Hint: Use the concept of Doppler shift for light to find the speed of the star. The formula of doppler shift is given by $\dfrac{{\Delta \lambda }}{\lambda } = \dfrac{v}{c}$. The percent decrement in the wavelength is given in the question which can directly be used in the formula.

Complete step by step answer:
The Doppler shift for light is given by the formula
$\dfrac{{\Delta \lambda }}{\lambda } = \dfrac{v}{c}$
Where,
$\Delta \lambda$= change in wavelength of light.
$\lambda$=actual wavelength of light.
v= velocity of the source.
c= velocity of light.
It is given in the question that,
The percent change in wavelength $= 0.05\%$
$\dfrac{{\Delta \lambda }}{\lambda }\left( {100} \right) = 0.05$
$\dfrac{{\Delta \lambda }}{\lambda } = \dfrac{5}{{{{10}^4}}}$
putting in the formula given earlier
$\dfrac{5}{{{{10}^4}}} = \dfrac{v}{{3 \times {{10}^8}}}$
$v = 1.5 \times {10^5}\dfrac{m}{s}$
We know that, when the source moves toward the observer then the apparent wavelength is less than the actual wavelength which is exactly the case given in the question. The wavelength of the source seems to be decreased by $0.05\%$which means that the source is coming toward the observer.
c) is correct.

Additional information:
When a light source moves toward or away from an observer, certain shifts in the value of wavelength and frequency of light are observed. If the source is going away from the observer in its direct line of sight, the wavelength of the light coming from the source will shift toward the red region of the visible light spectrum. This phenomenon is known as red shift. Similarly, if the source is moving toward the observer then the wavelength will shift toward the blue region and this phenomenon is called blue shift. The concept of redshift and blueshift is used to calculate the direction and speed of a distant star just by observing the light coming from it.

Note: We cannot use the Doppler shift formula of sound waves for light waves. So, a general question arises here: If both sound and light travel as waves then why does the formula differ for them?
This is because the Doppler shift for sound waves is derived by taking a medium (in which sound is travelling) to be at rest and both source and observer moves relative to this medium.
On the other hand, in the case of light, there is no such thing which is at universal rest. Everything is moving with respect to each other.